Overview
Calculating with decimals
Fractions and reciprocals
Fractions of quantities
Adding and subtracting
fractions
Multiplying and dividing
fractions
Fractions on a calculator

8

Linear graphs

7.1
7.2
7.3
7.4
7.5

8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8

63
65
67
69

71
73
75
77
79
81
83

Overview
85
Graphs and equations
87
Drawing linear graphs by
finding points
89
Gradient of a line
91
y = mx + c
93
Finding the equation of a line
from its graph
95
The equation of a parallel line 97
Real-life uses of graphs
99
Solving simultaneous
equations using graphs
101

Percentages and
compound measures
Overview
Equivalent percentages,
fractions and decimals
Calculating a percentage of a
quantity
Increasing and decreasing
quantities by a percentage
Expressing one quantity as a
percentage of another
Compound measures

Introduction
Welcome to Collins AQA GCSE Maths Foundation Teacher Pack, which has been written for
the 2015 AQA GCSE Mathematics Specification (8300). This Teacher Pack accompanies
Collins AQA GCSE Maths Foundation Student Book 4th edition, which contains the material
needed to complete the AQA course.
The new GCSE contains some types of question that have not appeared in GCSE
Mathematics exams before. This guide and its lesson plans will help you prepare students to
tackle the new aspects with confidence.
Aims and learning outcomes
In mathematics, topics are taught in progressively greater depth over the secondary phase.
GCSE outcomes may reflect or build on subject content knowledge that is typically taught at
KS3. There is no expectation that such content will be repeated during the GCSE course,
when it has already been taught effectively at an earlier stage. This allows for some increase
in content, with the Government recommendation that mathematics is taught for a minimum
of seven hours per fortnight.
This GCSE course is designed to provide a broad and coherent course of study that
encourages students to develop confidence in, and a positive attitude towards, mathematics.
It enables students to recognise the importance of mathematics in their own lives and to
society.
The aims and objectives of this GCSE Mathematics course are to enable students to:
• develop fluent knowledge, skills and understanding of mathematical methods and
concepts
• acquire, select and apply mathematical techniques to solve problems
• reason mathematically, make deductions and inferences, and draw conclusions
• comprehend, interpret and communicate mathematical information in a variety of
forms, appropriate to the information and context.
The new GCSE will have more focus on making sure that every student masters the
fundamentals of mathematics. These have been defined by the Department for Education as
areas such as calculation, ratio and proportion.
Key changes to GCSE Mathematics:
• new content in both Foundation and Higher tiers
• longer assessment time (four hours)
• additional content in the Foundation tier
• more formulae need to be known (only cone, sphere and kinematics are given)
• seven hours of study per fortnight is recommended.
Modified content
Some content:
• is a result of more knowledge assumed from earlier key stages, for example, knowledge
of up to the 12 times table from Key Stage 2
• is more explicitly stated but may or may not have been implied previously, for example,
‘expansion of more than two binomials’ and ‘unique factorisation theorem’.

Content added to Foundation and Higher
Some content is completely new to both tiers, for example:
• systematic listing strategies
• Fibonacci-type sequences
• quadratic sequences
• simple geometrical progression
• pressure
• functions
• frequency trees
• Venn diagrams.
Examples of content added to Foundation, previously only in the Higher Tier:
• calculate exactly using multiples of π
• standard form
• apply and interpret limits of accuracy
• expand the product of two binomials
• solve quadratic equations by factorisation
• solve linear simultaneous equations
• trigonometric ratios in 2D right-angled triangles
• fractional scale enlargements in transformations
• lengths of arcs and areas of sectors of circles
• vectors (but not proofs)
• density
• tree diagrams.
Examples of content no longer in the Foundation specification:
 Trial and improvement
 Tessellations
 Questionnaires and surveys
New Assessment Objectives
Assessment Objectives (AOs) are related to those skills that are linked to subject content,
but not specifically. In mathematics there were, and still are, three AOs. There are significant
changes in the focus of the AOs to bring a new challenge to the examination papers.
The DfE summarises these changes as follows:
‘The assessment objectives place more emphasis on reasoning and problem solving.’
First, the language of the AOs has changed:



AO1 ‘Recall and use their knowledge’ has become ‘Use and apply’
AO2 ‘Select, apply methods …’ is now ‘Reason, interpret and communicate’.

Both changes suggest a stronger emphasis on application rather than on recall.
Secondly, the revised version provides more guidance as to what they mean in practice. For
example, the old AO3 stated that students needed to ‘interpret and analyse problems’; the
new AO3 states that they need to ‘solve problems within mathematics and other contexts’.
This includes making connections between different areas of mathematics.
Finally, the weighting has changed, with more marks for AO3 than before, and a new
separation of Foundation and Higher weighting.

AO1 still includes recall and standard procedures, but has been extended to include
multi-step solutions and questions set in context.
AO2 is all about interpreting information and communicating solutions and arguments,
giving these areas much more emphasis than previously.
AO3 is still about solving problems, but the emphasis is on translating problems into
mathematical processes, linking different areas of mathematics and evaluating solutions.
Assessment Objectives

AO1

AO2

AO3

Use and apply standard techniques
Students should be able to:
• Accurately recall facts, terminology and definitions.
• Use and interpret notation correctly.
• Accurately carry out routine procedures of set tasks
requiring multi-step solutions.
Reason, interpret and communicate mathematically
Student should be able to:
• Make deductions, inferences and draw conclusion from
mathematical information.
• Construct chains of reasoning to achieve a given result.
• Interpret and communicate information accurately.
• Present arguments and proofs.
• Assess the validity of an argument and critically
evaluate a given way of presenting information.
Where problems require candidates to ‘use and apply
standard techniques’ or to independently ‘solve problems’,
attribute a proportion of those marks to the corresponding
Assessment Objective.
Solve problems within mathematics and in other
contexts
Students should be able to:
• Translate problems in mathematical or nonmathematical contexts into a process or a series of
mathematical processes.
• Make and use connections between different parts of
mathematics.
• Interpret results in the context of a given problem.
• Evaluate methods used and results obtained.
• Evaluate solutions to identify how they may have been
affected by assumptions made.
Where problems require candidates to ‘use and apply
standard techniques’ or to ‘reason, interpret and
communicate mathematically’, attribute a proportion of
those marks to the corresponding Assessment Objective.

How to use this book
Chapter overview
Each chapter starts with an outline of the content covered in the entire chapter, to help you
plan ahead with ease.
• Overview shows the topic in each lesson at a glance.
• Prior learning indicates the mathematical ideas with which students will need to be
secure, to enable them to access the topic content.
• Learning objectives list what students will learn.
• Extension suggests ways to broaden and deepen mathematical understanding with
appropriate challenge for higher-attaining students.
• Curriculum references show how the material meets the requirements of the new
GCSE curriculum with references to the KS3 and KS4 Programmes of Study.
• Route mapping chart for all exercises indicates, at a glance, the level of work students
will meet. Key questions are referenced. These require a step-up in understanding or
application and/or provide opportunities for students to demonstrate mastery of a
concept.
• About this chapter explains how the content in the chapter links mathematical ideas
and makes connections. The relevance to everyday life is highlighted, with references
to core skills and possible career foci. Some ideas for probing questions are included to
encourage students to work mathematically and suggestions are made for
assessment. In addition, there are suggestions on how to use the worked exemplars
from the Student Book.
Lesson plans
Every section in the Student Book is supported by a sequence of lesson plans. Each lesson
plan follows the same format, making them easy to use as an aid in preparing lessons.
•
•
•
•
•
•
•
•
•

Resources and homework provide the corresponding Student Book page numbers
and Practice Book references.
Learning objectives indicate clearly what the lesson is about and the level of the
content, providing a useful tool for measuring the success of a lesson.
Making mathematical connections identifies areas of mathematics that require similar
skills.
Making cross-curricular connections identifies how the skills and knowledge are used
in other areas of the curriculum.
Prior learning identifies the skills and knowledge that students will need in order to
access the lesson.
Working mathematically highlights and develops core skills that enable students to
work independently and explain and apply their mathematics.
Common misconceptions and remediation explores common errors that students
may make, with suggested approaches for identifying or avoiding the errors. It also
highlights possible areas of misunderstanding.
Probing questions offer some possible questions for teachers to ask of students to
encourage mathematical thinking and explanations, and to broaden and deepen their
understanding of the core ideas.
Literacy focus identifies the key terms and vocabulary. It also suggests written or
spoken activities, where appropriate.

A three-part lesson plan for each section of the chapter
Part 1, Part 2 and Part 3 structure the route through the lesson, with references to
Student Book exercises. This enables teachers to identify the core ideas to be taught
before students attempt the relevant exercises and activities. Part 1 is intended to be an
introductory activity, Part 2 is the main, application, part of the lesson and Part 3 is the
plenary or assessment opportunity.

Answers to questions
• Answers to the exercises and questions that appear in the Student Book are available
online at www.collins.co.uk/gcsemaths4eanswers or on the CD-ROM that
accompanies this Teacher Pack.
• Answers to the Practice Book questions can be found online at
www.collins.co.uk/gcsemaths4eanswers
Schemes of Work
Flexible 2-, 3- and 5-year Schemes of Work are available at the front of this Teacher Pack
and also on the CD-ROM. These are based on the Collins Maths Frameworking 3rd edition
Pupil Books 1.1, 1.2, 2.1, 2.2, 3.1 and 3.2 plus the AQA GCSE Maths Foundation Student
Book, 4th edition.
CD-ROM
The CD-ROM contains the entire Teacher Pack material in Word, to enable you to customise
lessons.
It also includes:
• Activities (quick starter activities and extension activities)
• Literacy activities
• 2-, 3- and 5-year Schemes of Work
• Answers to the Student Book questions.

Chapter 1 Number: Basic number
Overview
1.1 Place value and order numbers

1.3 The four rules

1.2 Order of operations and BIDMAS
Prior learning
Know the multiplication tables up to 12 × 12 and associated division facts.
Know squares of numbers, up to 15 × 15 = 225 and the square roots of the square numbers
up to 225.
Know the addition and subtraction of numbers less than 20.
Know how to multiply numbers by 10 and 100.
Know what a fraction and a negative number represent.
Learning objectives
By the end of this chapter, ensure that students can: compare and order positive and
negative numbers; use the order of operations when calculating; add, subtract,
multiply and divide positive and negative integers and positive decimals without a
calculator.
In the examination, students will be expected to:
•
use a number line to represent negative numbers
•
use inequalities with negative numbers
•
compare and order positive and negative numbers
•
work out to rules of arithmetic with integers and decimals.
Extension
Ask students to write problem-solving questions based on real-life situations.

Key questions are those that demonstrate mastery of the concept, or which require a step-up
in understanding or application. Key questions could be used to identify the questions that
students must tackle, to support differentiation, or to identify the questions that should be
teacher-marked rather than student-marked.
About this chapter
Making connections: This chapter teaches students how to explain an easy way to do
multiplications and divisions mentally, and why the knowledge of factors will help them with
this. It also teaches students why we use brackets in some calculations, and the advantages
of using inequality signs.
Relevance: Linking questions into real-life problems involving the four rules, including
negative numbers.
Working mathematically: Students question what to look for when deciding if they can do a
calculation mentally. Students explain the steps of the calculation, and write the steps in
order. Students think about what happens when they subtract a negative number from
another negative number.
Assessment: In each section of this chapter, ensure that students have a good grasp of the
key questions in each exercise before moving on. (Refer to the ‘Route mapping’ table.)
Encourage students to read and think about the ‘Ready to progress?’ statements on page 32
of the Student Book. Check students’ understanding at the end of the chapter, formatively,
using peer assessment. Students could do a mini test in the form of the ‘Review questions’
on pages 32–33 of the Student Book. Follow up the test with an individual target-getting
session, based on any areas for development that a student may have.
Worked exemplars from the Student Book (page 31) – suggestions for use
•
Present students with the same question but different numbers. They should use the
exemplar to mirror the working, in full or just refer to the notes.
•
Copy and cut the exemplars into cards. Students match the working with the notes.
•
Alternatively, copy and cut the working into cards but split the label/description from the
working.
Answers to the Student Book questions are available on the CD-ROM provided.

Use a number line to represent negative •
numbers
•
Use inequalities with negative numbers
Compare and order positive and negative
numbers

Student Book 1.1: pages 9–11
Practice Book 1.1

Making mathematical connections

Making cross-curricular connections

•
•

•

Multiplying and dividing by powers of 10
Multiplying and dividing decimals

•

Science – place value in calculations;
ordering a set of results from an
experiment
Relevance – students must be able to
understand these basic concepts

Prior learning
•

•

Students should know the size order of the words units, tens, hundreds, thousands,
ten thousands, hundred thousands and million, and understand the relative values of
the terms.
Students could sort number cards into order as a test of this knowledge.

Working mathematically
•
•

Structure tasks so students can work out the methods for themselves, either by increasing
the difficulty incrementally or through one straightforward and one complex example.
Ask students to describe a set of numbers using inequalities.

Common misconceptions and remediation
•
•

Students may find it difficult to identify and say very large numbers, particularly those
that include zeros, for example: 70 403. (Seventy thousand, four hundred and three)
Students will sometimes confuse inequality signs. Encourage them to remember: The
crocodile eats the largest number. Or, students may prefer to remember that the
arrow points to the smaller number.

Probing questions
•
•
•

How would you order this set of numbers?
Is –9 bigger or smaller than –2?
How do you know that one number is bigger than another?

Literacy focus
•
•

Key terms: digit, positive, inequality, place value, negative
Ask students to write some of the larger numbers in Exercise 1A in words.

Part 1
•
•

Remind students that numbers can carry on to infinity – however large a number they think
of, they can always add one more.
It is the number and order of digits in a number, not the value of the individual digits, which
determines the size of a number. Hold up some digit cards. Shuffle them and place them
AQA GCSE Maths
Foundation Teacher Pack – Chapter 1

face down. Ask students to draw four empty boxes next to each other. Say that you will pick
four numbers at random, and they should write them down, one at a time, deciding where to
place each digit in the boxes to make the biggest four-digit number possible.
Play this several times, giving a point to each student who makes the biggest number each
time.
Extend this by asking students to find the biggest odd number or the biggest even number.

Part 2
•
•
•

•

•

•
•
•

•

The number line
Remind students of place values – units, tens, hundreds, thousands, millions, and so on.
Less able students may wish to continue using the place-value cards.
Show students a four-digit number and ask for the value of each digit; for example, display
the number 3657 and ask for the value of the 6 (600). Repeat for larger numbers.
Point out that when writing numbers with more than four digits, the convention is to leave
narrow spaces between groups of three digits, starting from the right. (Commas are not used
as separators, to avoid confusion, because they are used as the decimal marker in many
countries.) These spaces help to identify the number. For example, 2 678 000 is
two million, six hundred and seventy-eight thousand.
Write the number 3 001 002 on the board and ask students to read it aloud. Demonstrate the
importance of zeros – in this example, the number would become 312 (three hundred and
twelve) if the zeros were left out.
Ask students to use the numbers 1, 2 and 3 (once each) to make as many three-digit
numbers as possible. They should then sort them into order, from the smallest to the largest
(123, 132, 213, 231, 312, 321).
Take students through Example 1 to consolidate their understanding of place values.
Now go through inequality signs with students explaining their uses.
A more able class may finish Exercise 1A quite quickly. If this is the case, ask students to
work in pairs to make up their own questions on place values and ordering, taking turns to
create a question and answer it.
Students can now do Exercise 1A from the Student Book.
N 1–3, 8, 10 Calculator n/a CM 12, 13
MR 7, 9, 11
PS 4–6, 14
EV n/a

Part 3
•
•

Read out some numbers such as 109, 7043 or 4508, and ask students to write them down in
figures.
If there is time, ask students to make up a number with a zero in one of the places and read
it out for the rest of the class to write down.

Students should know the number bonds to 20 and the multiplication tables up to 10 ×
10.

Working mathematically
•

By the end of this section, make sure that students can confidently apply BIDMAS/BODMAS
to calculations.

Common misconceptions and remediation
•
•

Some students ignore the conventions in the order of operations and work from left to
right; for example, they may work out the answer to 4 5 × 6 as 54 rather than 34.
Encourage students to recognise questions on this topic and to work in the correct
order. As a reminder before doing the calculation they should write down the
mnemonic BIDMAS/ BODMAS (Brackets; Indices/Powers; Division; Multiplication;
Addition; Subtraction).

Probing questions
•
•

Where would you place brackets in order to make this calculation correct?
What is the answer to this calculation: 4 + 3 × 6? Why?

Literacy focus
•
•

Key terms: None in this section
From memory, ask students to write down the words for the mnemonic BIDMAS as quickly
as they can.

Part 1
•
•
•

•
•

Ask students to work out:
4 ÷ 4 + 4 – 4 (= 1)
(4 + 4) ÷ (4 + 4) (= 1)
(4 ÷ 4) × (4 ÷ 4) (= 1)
Make sure that students understand that the parts in brackets must be done first.
Ask students to make any number other than 1, using only four 4s. For example:
(4 ÷ 4) + (4 ÷ 4) = 2
(4 + 4 + 4) ÷ 4 = 3
(4 – 4) ÷ 4 + 4 = 4
(4 × 4 + 4) ÷ 4) = 5
Ask students to feed back their calculations to the class and take time to explain any
calculations that need adjusting in order to use brackets correctly.
Give students four different numbers (e.g. 1, 2, 3, 4) and ask them to use these numbers to
make 21, e.g.: (1 + 2) × (3 + 4). Tell students that all these calculations require them to use
the operations in the correct order.
AQA GCSE Maths
Foundation Teacher Pack – Chapter 1

Write 2 3 × 4 on the board. Ask students to work out their answers on a blank sheet of
paper. Write the numbers 14 and 20 on the board and collect students’ answer sheets.
Write a tally under the numbers 14 and 20 to show how many students got each answer.
Explain that 14 is the correct answer because the convention is that multiplication is worked
out before addition, so 2 + 3 × 4 = 2 + 12. Say that there is a convention for the order of
carrying out operations in calculations and they must follow it. Refer students to the
BIDMAS/BODMAS list in the Student Book. Explain that all calculators are programmed to
use BIDMAS/BODMAS.
Explain that, if a calculation has brackets in it, that part must be worked out first, and so on.
Write 2 + 3 × 4 and (2 + 3) × 4 on the board. Remind students that the answer to the first
calculation is 14. Ask students for the answer to the second calculation (20).
Write (1 + 2 + 3) × 4 and ask students to give the answer (24).
Explain that sometimes brackets are not needed; for example, 3 × 4 + 2 is just as
mathematically correct as (3 × 4) + 2 because 3 × 4 will be the first calculation in both cases.
Ask students to work in pairs, using the numbers 1, 2, 3 and 4 and a combination of
brackets, division, multiplication, addition and subtraction, to create as many different
answers as possible. Allow them to combine numbers, e.g. 1 and 2 to be used as 12.
For less able students, limit the number of numbers they use. If the class is more able, give
them extra rules that they must take into account when making their calculations, e.g. using
four numbers from 1 to 20 only once to create as many answers as possible.
Now go through Example 2 in the Student Book with the class.
Students can now do Exercise 1B from the Student Book.
N 1–6, 9, 14 Calculator n/a CM 8, 10
MR 7, 13, 16 PS 11, 12, 15
EV n/a

Part 3
•
•
•

Play a game of ‘Countdown’, choosing a set of numbers such as: 25, 6, 3, 10, 4, 1.
Give students 30 seconds to make, for example, 254 (e.g. 25 × 10 + 4) or 60 seconds to
make, for example, 287 (25 × 10 (6 3) × 4 1).
Pick other target numbers of varying difficulty, depending on the ability of the class.

Students should know the number bonds to 10.
Students should know the multiplication tables up to 10 ×10, including calculations
with zero.

Working mathematically
•
•

Students may find it helpful when copying out questions for addition and subtraction
to set the digits into columns (as in Example 3 of the Student Book).
Encourage students to use their preferred method for calculations.

Common misconceptions and remediation
•

•
•
•

When subtracting, students may take the smaller digit from the larger in a column rather than
‘bottom from top’. This revisits earlier teaching methods of subtracting by taking smaller from
larger. Emphasise the correct method and explain how to ‘borrow’ numbers.
When multiplying and dividing, students may ignore zeros, or forget to add any carried digits.
Make students aware of these errors and encourage them to check their work
carefully.
Make sure that pupils overcome the confusion about ‘two negatives make a positive’, for
example: 6 – –9 = 6 + 9 or –6 × –9 = 54.

Probing questions
•

Before starting each exercise, give students questions with errors and say: Spot the errors.

Literacy focus
•
•

Key terms: column method, grid method, partition method
Ask students to write a step-by-step guide for each of their preferred calculation methods.

Part 1
•

Give students a two-digit number. Say: What number must we add to make 100? Repeat,
making the numbers more difficult. Next, ask students questions such as: If 7 × 10 is 70,
what is 70 × 10? Extend this by giving answers to multiplications; ask for possible questions.

Part 2
•

Addition with positive numbers; Subtraction with positive numbers
Work through Example 3 with the class. Say: the answer will always be larger than the larger
number; add the units column first; when the digits total in a column is more than 9, write the
carried digit under the next column on the left so that you do not forget to add it in.
AQA GCSE Maths
Foundation Teacher Pack – Chapter 1

Work through Example 4. Say: the answer will be smaller than the larger number; subtract
the units column first; ‘decompose’ a number from the next column to the left, if necessary.
Multiplication with positive numbers; Division with positive numbers
Work through Example 5. Say: ‘When multiplying, write the bigger number first: multiply the
units. When working with positive numbers larger than 1, expect a bigger answer than the
numbers you began with’.
Students may prefer to use a grid method. Say that this is acceptable but show students that
these methods do the same thing, e.g. 14 × 4 = 10 × 4 + 4 × 4 (10 fours and 4 fours).
Work through Example 6. Point out that in division (when working with numbers that are
larger than 1) the answer will be smaller.
Students can now do Exercise 1C from the Student Book.
N 1–5, 9–11, 14, 15
Calculator n/a
CM n/a MR 16
PS 8, 12, 13
EV 6,7
Arithmetic with negative numbers
Work through the Student Book text and Examples 7 and 8. Provide a variety of addition and
subtraction questions involving negative numbers using the thermometer or a number scale.
Students can now do Exercise 1D from the Student Book.
N 1 –3, 5 Calculator n/a
CM n/a
MR 4
PS 6, 7
EV n/a
Adding and subtracting negative numbers
Work through Examples 9 and 10 in the Student Book with the class.
Students can now do Exercise 1E from the Student Book.
N 1–10, 12, 13 Calculator 10 CM 11
MR n/a
PS 14–17
EV n/a
Multiplying and dividing with negative numbers
Work through the text and Example 11 in the Student Book with the class.
Students can now do Exercise 1F from the Student Book.
N 1, 2, 5–7, 10, 11a, b, 13 Calculator n/a
CM 11c
MR 3, 4
PS 8, 9 EV 12
Long multiplication with integers
Work through the text and Examples 12, 13 and 14 with the class. If
there is a preferred method that your class likes to use, do a few more
examples using this method.
Students can now do Exercise 1G from the Student Book.
N 1, 2, 4 Calculator n/a CM 7 MR 3 PS 5, 6 EV n/a
Long division
Work through the text and Examples 15, 16 and 17 with the class.
Some students will still use a short division method, which you should
allow if their answers are accurate.
Students can now do Exercise 1H from the Student Book.
N 1–3, 7
Calculator n/a
CM n/a
MR 6, 9
PS 4, 5, 8
EV n/a
Long multiplication with decimals
Work through Example 18 with the class. Another method is to ignore all decimal places and
multiply using students’ preferred method. Then students simply count how many digits there
are after the decimal points and use place value to insert the decimal point.
Students can now do Exercise 1I from the Student Book.
N 1, 2, 4
Calculator n/a
CM n/a
MR n/a
PS 3, 5
EV n/a